Three-cycle forces a two-iterate interval horseshoe

For a continuous interval map $F$ with a three-cycle $a<b<c$, after reflecting the coordinate if needed its ordering is $F(a)=b$, $F(b)=c$, $F(c)=a$. Then $F^2(a)=c$, $F^2(b)=a$. Also the intermediate value theorem gives $d\in (b,c)$ with $F(d)=b$, hence $F^2(d)=c$. Each of $[a,b]$ and $[b,d]$ therefore maps across $[a,c]$ under $F^2$. Choose first-passage subintervals between endpoint levels to avoid overshoots; their interiors are disjoint and both map exactly onto $(a,c)$. This is a horseshoe for an interval map, and establishes Glendinning chaos.